Homogeneous Linear Equations with More Unknowns than Equations

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Theorem

Let $\alpha_{ij}$ be elements of a field $F$, where $1 \le i \le m, 1 \le j \le n$.

Let $n > m$.

Then there exist $x_1, x_2, \ldots, x_n \in F$ not all zero, such that:

$\ds \forall i: 1 \le i \le m: \sum_{j \mathop = 1}^n \alpha_{ij} x_j = 0$


Alternatively, this can be expressed as:

If $n > m$, the following system of homogeneous linear equations:

\(\ds 0\) \(=\) \(\ds \alpha_{11} x_1 + \alpha_{12} x_2 + \cdots + \alpha_{1n} x_n\)
\(\ds 0\) \(=\) \(\ds \alpha_{21} x_1 + \alpha_{22} x_2 + \cdots + \alpha_{2n} x_n\)
\(\ds \) \(\cdots\) \(\ds \)
\(\ds 0\) \(=\) \(\ds \alpha_{m1} x_1 + \alpha_{m2} x_2 + \cdots + \alpha_{mn} x_n\)

has at least one solution such that not all of $x_1, \ldots, x_n$ is zero.


Proof


Sources